Generators of maximal left ideals in Banach algebras

TL;DR

This paper provides a shorter proof and extensions of a theorem relating finite-dimensionality of Banach algebras to properties of their maximal ideals, building on classical results from the 1970s.

Contribution

It offers a more concise proof of Ferreira and Tomassini's result and presents new extensions and examples related to maximal ideals in Banach algebras.

Findings

Shorter proof of the maximal ideal finite-dimensionality theorem

New extensions of the classical results

Additional examples illustrating the theory

Abstract

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over $\C$ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the \v{S}ilov boundary of $A$'. We shall give a shorter proof of this latter result, together with some extensions and related examples.